# RUN-104 — DAG-A in characteristic zero: Castella–Sano's determinantal Kato main conjecture read object by object and instantiated at `(389.a1, 11)` — it holds by cited theorem, so a unique `Λ`-basis `𝔷_{Q∞}` of `det⁻¹_Λ RΓ(Z_S,𝐓)` exists; specialised to the trivial character it is a `Z₁₁`-basis of `det H¹ ⊗ det⁻¹ H² = Z₁₁(P∧Q) ⊗ Sel_str(T)` — the rank-2 determinant tensored with the strict-Selmer line, which is where the element lives because `(str-fin)` fails there — and the core-vertex criterion of Macias Castillo–Sano's Definition 2.16, evaluated at the tree's own vertex `397·991`, gives core rank **1**, not 2, so the Stark image `ε₁` is a basis of a rank-one module and nothing is forced: **A PASS by cited theorem; A-mod11-compat UNMEASURED and not necessary**

**Arm:** 數學戰士「墜衡」 / AMRAL Research Lab
**Subject:** the symbolic line's instruction after [RUN-103](../RUN-103/) — keep RUN-103; read Conjecture 2.2.2's objects literally; Theorem A and Theorem B with their hypotheses re-instantiated; ideal level versus line level versus distinguished element; MCS Theorem 2.20 / 1.1, Theorem 3.4 / 1.5, Remark 3.14 and the Stark-module definitions, with ring, core rank, degree, dual, Selmer structure, integral lattice and hypotheses itemised at `T = T₁₁(389.a1)`; compose `z_det^Kato ↦ ε_Stark^Kato =: Δ_A`; mod-11 compatibility not required for A; compute the core rank and do not force; ten mutations, two positive controls; four outcomes; D and G closed
**Tools:** `src106_dag_a_char0_determinantal_kato.py`, `src11_gate_drill.py`
**Logs:** `src106-dag-a-char0-determinantal-kato.json`, `src11-gate-drill.json`
**Deliverable:** [`handoff/DAG-A-CHAR0-DETERMINANTAL-KATO/`](../../packages/DAG-A-CHAR0-DETERMINANTAL-KATO/) · **the papers' statements:** `data/external/literature/bockstein-and-kurihara-2026-09-20/EXTRACTS.md` (RUN-104 section)

**Result. Castella–Sano §2.2–2.3 and §1.1.4 and Macias Castillo–Sano §2–3 were read this round and their statements recorded verbatim. Conjecture 2.2.2 has eight objects: the ring `Λ = Z₁₁[[Γ]]`; the module `𝐓 = T ⊗ Λ` over `Z_S = Z[1/(11·389)]`; the perfect complex `RΓ(Z_S,𝐓)` of Euler characteristic `−1`; the hypothesis that `H²(Z_S,𝐓)` is torsion (Kato 12.4(1)); the invertible module `det⁻¹_Λ RΓ(Z_S,𝐓)`; the canonical map (2.9) to `Q(Λ) ⊗ H¹`, whose integral form is injective exactly when `H²` is torsion (Remark 2.2.1); the element `z_∞^{(S)} = 𝒫₃₈₉·z_∞` with `𝒫₃₈₉ = 1 − 389⁻¹γ₃₈₉` (`a₃₈₉ = +1` recomputed: split multiplicative; at the trivial character the factor is `388/389`, an 11-adic unit); and the statement that the preimage of `z^{(S)}` is a `Λ`-basis. Three levels are kept apart: the ideal equality (2.12), the line equality, the distinguished element — Proposition 2.2.3 makes the first equivalent to the other two together, and only the last two drive A. The main conjecture at `(389.a1, 11)` holds by cited theorem: CS's proof of Theorem B routes the good-ordinary case through the Mazur–Swinnerton-Dyer conjecture proved in Wan15 and BCS25, Kato's equivalence with Conjecture 12.10, and Proposition 2.2.3; the hypotheses CS list — `p > 3`, (sur), good ordinary, (tor_Q) — are instantiated from the tree (`#E(F₁₁) = 16`, `a₁₁ = −4` recomputed; RUN-069's surjectivity witnesses; trivial torsion); `p ∤ c_φ` is needed only for Kurihara's Conjecture B and is not used. So `𝔷_{Q∞}` exists, is unique, and is nonzero. At the trivial character CS's own sequence (2.13), with every local `H²` vanishing (`E(Q₁₁)[11^∞] = 0`, `E(Q₃₈₉)[11^∞] = 0`, both computed), gives `H²(Z_S,T) ≅ Sel_str(E[11^∞])^∨` unconditionally, hence `rank H¹ = 1 + rank H²`; the formal-group depth of `16P` and `16Q` at 11 (both in `E₁∖E₂`; `u_P = 9`, `u_Q = 6`, additive) shows `loc₁₁` is onto with index `11⁰`, so under the tree's inherited `Ш[11] = 0` the ranks are `(2, 1)` — by two independent routes — `H²` is torsion-free, and `det⁻¹RΓ(Z_S,T) = det H¹ ⊗ det⁻¹ H² = Z₁₁(P∧Q) ⊗ Sel_str(T)`, the strict line being `F₁₁·(P + 4Q)` mod 11. `(str-fin)` fails (`Sel_str` has corank 1), so CS's descent map does not exist at `α = 1`: the distinguished element does not descend to `H¹`; its placement is the rank-2 determinant tensored with the strict-Selmer line, degrees `[2, −1]`. The core rank of the relaxed structure — the one CS's complex carries — is `1` by Proposition 2.4(ii) (`rank H⁰(R,T^*(1)) = 1`, `T⁻ = 0`) and `1` again by Definition 2.16's formula (2.2.2) at the tree's own vertex `397·991` on `E[11]`: `dim H¹_{𝓕^n} = 3`, local dimensions `1, 1`, the third core-vertex condition being exactly RUN-022's `Sel_both = 0`, i.e. `det M_loc = 2 ≠ 0`; the ordinary structure has core rank `0` (Remark 3.14's `SS₀`). Hypothesis 2.11 (i)–(iv) is verified for the relaxed structure, `RΓ_{/ur}(Q₃₈₉,T)` is acyclic so MCS's Selmer complex IS `RΓ(Z_S,T)`, and Theorem 3.4 gives the canonical `ϖ₁ : det⁻¹RΓ(Z_S,T)/11 ≅ SS₁(E[11], 𝓕_can) = ∧³H¹_{𝓕^n}(Q,E[11]) ⊗ W₃₉₇ ⊗ W₉₉₁`; `ε₁ = ϖ₁(𝔷_{Q∞}(1))` is a basis, hence `Δ_A ≠ 0` at once; Corollary 3.5 turns it into Fitting ideals of the STRICT Selmer group (`I₀ = 0`, `I₁ = F₁₁`). The Selmer structure is constant along the whole chain; the ordinary channel (Remark 3.14, `Θ₂ = Z₁₁`, RUN-068's `δ̃ = 5` a level-2 witness) is read in parallel and never composed with it. `B_N`'s line shares the local factor `W₃₉₇ ⊗ W₉₉₁` with the Stark line and differs in the global one (`(∧²Sel)⁻¹` against `∧³H¹_{𝓕^n}`); what `det(loc_n) ≠ 0` does is certify the core vertex — an equality of elements is UNMEASURED and, by the instruction, not necessary. Verdict A PASS by cited theorem. The ten mutations and both controls are planted; D and G were not opened.**

---

## 0. The verdict in the requested form

```text
A = PASS   (route: CITED_THEOREM)
   KATO-DETERMINANT-IMC (Conjecture 2.2.2 at (389.a1, 11))   = PASS / CITED_THEOREM — Wan15 + BCS25 via CS Theorem B's proof, Kato's
                                                                equivalence, CS Prop 2.2.3; hypotheses p > 3, (sur), good ordinary, (tor_Q)
                                                                instantiated in the tree (§2); UNMEASURED: the internal hypotheses of the
                                                                cited main-conjecture papers, Kato 12.4(1) (read through CS), optimality
   canonical determinant line                                = PASS — det⁻¹_Λ RΓ(Z_S,𝐓), no choice (§1)
   distinguished element 𝔷_{Q∞}                               = PASS — the unique preimage of z^{(S)} (Remark 2.2.1), a basis, nonzero (§1–2)
   core rank χ(𝓕_can)                                         = 1 (two formulas, §4) — NOT 2; nothing forced
   placement at the trivial character                        = 𝔷_{Q∞}(1) is a Z₁₁-basis of det H¹ ⊗ det⁻¹ H² = Z₁₁(P∧Q) ⊗ Sel_str(T),
                                                                degrees [2, −1]; no descent to H¹ since (str-fin) fails (§3)
   Stark image ε₁ = ϖ₁(𝔷_{Q∞}(1))                              = PASS — a basis of SS₁(E[11], 𝓕_can, 𝒫₁) ≅ ∧³H¹_{𝓕^{397·991}} ⊗ W₃₉₇ ⊗ W₉₉₁;
                                                                Theorem 3.4 cited, Hyp 2.11 (i)–(iv) and Hyp 2.17 (at n = 1) verified (§5)
   A-char0                                                    = PASS / CITED_THEOREM
   A-mod11-compat (ε₁ against [det loc_{397·991}])            = UNMEASURED — not necessary for A; det(loc_n) ≠ 0 is the core-vertex certificate
   scalar                                                     = UNMEASURED — the unit ε with 𝔷_{Q∞}(1) = ε·(P∧Q)⊗R_str is not computed;
                                                                Kim's u is a mod-11 quantity and is not used as it
   inherited input                                            = Ш(E/Q)[11] = 0 (P5 chain; never verified here) — needed only for the
                                                                NUMBERS 2, 1 and for the core-vertex certificate, not for the shapes
   B–G                                                        = untouched; D and G not opened
```

## 1. The objects of Conjecture 2.2.2, literally

| object | in the paper | here | status |
| --- | --- | --- | --- |
| coefficient ring | `Λ = Z_p[[Γ]]`, `Γ = Gal(Q_∞/Q)`; `Q(Λ)` its fraction field | `Z₁₁[[Γ]]` | — |
| Galois module | `𝐓`, the cyclotomic deformation of `T = T_p(E)`; `S = {∞} ∪ {ℓ ∣ Np}` | `T = T₁₁(389.a1)`, `S = {∞, 11, 389}`, `Z_S = Z[1/4279]` | — |
| complex | `RΓ(Z_S,𝐓)`, perfect, acyclic outside degrees 1, 2 [FK06] | the same; Euler characteristic `−rank T⁻ = −1` | CITED |
| torsion hypothesis | `H²(Z_S,𝐓)` is `Λ`-torsion [Kato Thm 12.4(1)] | a hypothesis of the conjecture; the reason (2.9) exists | CITED_THEOREM |
| determinant line | `det⁻¹_Λ RΓ(Z_S,𝐓)`, invertible | canonical: the determinant of a perfect complex needs no choice | PASS |
| the map (2.9) | `Q(Λ) ⊗ det⁻¹RΓ ≅ Q(Λ) ⊗ H¹`; Remark 2.2.1: an integral `det⁻¹RΓ → H¹` [BS21 2.18], injective iff `H²` torsion | injective here, so a preimage of an element of `H¹`, when it exists, is unique | PASS |
| distinguished Kato element | `z_∞^{(S)} = 𝒫_N z_∞`, `𝒫_ℓ = (1−α_ℓℓ⁻¹γ_ℓ)(1−β_ℓℓ⁻¹γ_ℓ)` | `a₃₈₉ = +1` **computed** (`#E_ns(F₃₈₉) = 388`): split, `(α,β) = (1,0)`, `𝒫₃₈₉ = 1 − 389⁻¹γ₃₈₉`; at the trivial character `388/389`, `v₁₁ = 0` | canonical |
| the statement | ∃ a `Λ`-basis `𝔷_{Q∞} ∈ det⁻¹RΓ` mapping to `z^{(S)}` under (2.9) | a line equality (the lattice `det⁻¹RΓ` equals `Λ·(2.9)⁻¹(z^{(S)})`) together with an element (that preimage, unique) | — |

**Three levels.** L1, the ideal equality (2.12) `char(H¹/Λz^{(S)}) = char(H²)`; L2, the line equality; L3, the distinguished element. Proposition 2.2.3: L1 ⟺ (L2 ∧ L3). Remark 2.2.4: the divisibility `⊂` alone ⟺ the preimage is integral. An ideal names no element — over `Λ` the ambiguity is `Λ^×` — so L1 read as producing the element by itself is planted and caught (`dag-a-objects`); so is "canonical" stretched from *the* element determined by `z^{(S)}` to every element of the line. Only L2 and L3 drive A.

## 2. Theorem A and Theorem B, with the hypotheses re-instantiated

Theorem A (`p > 3`, (sur), `p ∤ c_φ`): Kurihara's `M_∞(δ) = ord_p(Tam_E)` ⟺ Conjecture 2.2.2. Theorem B: under the same hypotheses Conjecture B holds when `p` is good ordinary, **because** — its proof, quoted in EXTRACTS — "the Iwasawa Main Conjecture for `Q_∞/Q` in the formulation of Mazur–Swinnerton-Dyer is proved in [48, 7]" (= Wan15, BCS25; the numbers resolved from CS's bibliography) and "each of these main conjectures is known to be equivalent to [24, Conj. 12.10] and hence to the Iwasawa Main Conjecture 2.2.2". What this round needs is the middle of that proof, not its conclusion:

| hypothesis | at `(389.a1, 11)` | label |
| --- | --- | --- |
| `p > 3` | 11 | COMPUTED_HERE |
| (sur) | RUN-069's six witnesses | DERIVED_FROM_VERIFIED_INPUTS |
| good ordinary at 11 | `#E(F₁₁) = 16`, `a₁₁ = −4`, `11 ∤ a₁₁` — recomputed | COMPUTED_HERE |
| (tor_Q) `E(Q)[11] = 0` | torsion trivial (RUN-011); also implied by (sur) | DERIVED_FROM_VERIFIED_INPUTS |
| `H²(Z_S,𝐓)` torsion | Kato Thm 12.4(1) | CITED_THEOREM |
| the MSD main conjecture | Wan15, BCS25 as invoked by CS; their internal hypotheses **not re-read** | CITED_THEOREM |
| Kato 12.10 ⟺ MSD; Prop 2.2.3 | CS §1.1.4, §2.2 | CITED_THEOREM |
| `p ∤ c_φ` | not needed for the main conjecture; Mazur78 Cor 4.1 would give it for the strong Weil curve since `11² ∤ 389`, optimality being external | EXTERNAL, unused |

The Skinner–Urban-type ramified prime is also present — `Δ = 389`, `v₃₈₉(Δ) = 1`, so `ρ̄₁₁` is ramified at 389 — but CS's route does not list it and this round does not lean on it. **KATO-DETERMINANT-IMC = PASS / CITED_THEOREM.** A theorem applied while a hypothesis is recorded as failing, or `a₁₁` misread as `0`, is planted and caught (`dag-a-imc`).

## 3. The trivial character: ranks, and where the element lives

CS's (2.13) at `α = 1`: `0 → H¹_rel(Q,T) → H¹(Z_S,T) → ⊕_{ℓ∣N} H¹(Q_ℓ,T)/H¹_f → Sel_str(E[11^∞])^∨ → H²(Z_S,T) → ⊕_{ℓ∣Np} H²(Q_ℓ,T) → H⁰(Q,T^*)^∨ → 0`. The `ℓ = 389` quotient is 0 (CS: `H¹_f = H¹_tor = H¹` for `ℓ ∣ N`), `H⁰(Q,T^*) = 0` by (tor_Q), and `H²(Q_ℓ,T) ≅ E(Q_ℓ)[11^∞]^∨` vanishes at both places: `11 ∤ 16` with `e = 1` at 11; `c₃₈₉ = 1` and `11 ∤ 388 = #F₃₈₉^×` at 389 (computed). Hence, unconditionally, `H²(Z_S,T) ≅ Sel_str(E[11^∞])^∨` and `rank H¹ = 1 + rank H²` (Euler characteristic `−1`, `H⁰ = 0`).

**The local image at 11.** `16P` and `16Q` lie in `E₁(Q₁₁) ∖ E₂(Q₁₁)` (`v₁₁(x) = −2` for both); the class `t/11 mod 11` on `E₁/E₂ ≅ F₁₁` gives `u_P = 9`, `u_Q = 6`, and it is additive (`P+Q ↦ 4`, `P−Q ↦ 3`, `2P ↦ 7` — checked). So `loc₁₁ : E(Q)⊗Z₁₁ → E(Q₁₁)⊗̂Z₁₁ ≅ Z₁₁` is onto (index `11⁰`), `loc₁₁ mod 11` has rank `ρ = 1`, and its kernel line — the strict-Selmer line mod 11 — is `F₁₁·(6P + 2Q) = F₁₁·(P + 4Q)`. Changing the multiplier 16 to another multiple prime to 11 (32, 48: drill controls) leaves the line and the index unchanged.

**The numbers, under the inherited input.** `Ш(E/Q)[11] = 0` is carried by the P5 chain (RUN-011, RUN-022) and has never been verified by this line; it is used here, labelled, for the numbers only: `corank Sel_{11^∞} = 2`, so `corank Sel_str = 1` (`loc₁₁` is onto a corank-1 divisible group), `rank H² = 1`, `rank H¹ = 2`, `H²` torsion-free (index `11⁰`), `H¹(Z_S,T) = Sel(T) = Z₁₁P ⊕ Z₁₁Q` (the quotient `H¹/Sel` injects into the torsion-free `H¹_s(Q₁₁,T)` with rank 0). The second route agrees: the image of `H¹(Z_S,V)` in the 2-dimensional `H¹(Q₁₁,V)` is its own annihilator, so 1-dimensional, and it contains the nonzero `loc₁₁(Sel(V)) = H¹_f(Q₁₁,V)`; hence `H¹(Z_S,V) = Sel(V)`.

**The line.** With CS's convention `det⁻¹RΓ = det(H¹) ⊗ det⁻¹(H²)`:

```text
det⁻¹_{Z₁₁} RΓ(Z_S,T)  =  Z₁₁·(P ∧ Q)  ⊗  Hom(H², Z₁₁)
                       =  Z₁₁·(P ∧ Q)  ⊗  T₁₁(Sel_str(E[11^∞]))  =  Z₁₁·(P ∧ Q) ⊗ Sel_str(T),   Sel_str(T) = Z₁₁·R_str,  R_str ≡ P + 4Q (mod 11)
```

and `𝔷_{Q∞}(1)`, the image of `𝔷_{Q∞}` under (2.14) (`det⁻¹_Λ RΓ ⊗_{Λ,1} Z₁₁ ≅ det⁻¹_{Z₁₁} RΓ(Z_S,T)`, FK06 1.6.5(3)), is a basis of it: `𝔷_{Q∞}(1) = ε·(P∧Q) ⊗ R_str` with `ε ∈ Z₁₁^×` **not computed**. `(str-fin)` fails, so CS's `ϑ` and Proposition 2.3.1 are not available at `α = 1`: **the distinguished element has no image in `H¹(Z_S,T)`; its home is the rank-2 determinant tensored with the strict-Selmer line, degrees `[2, −1]`.** This is the placement the instruction asked for instead of a forced identification. Building the characteristic-zero element backwards from `det loc_n = 2`, or taking Kim's `u ∈ (Z_p/I_n)^×` as the scalar `ε`, is planted and caught (`dag-a-placement`).

## 4. The core rank, two ways — it is 1

Proposition 2.4(ii): `χ(𝓕) = rank H⁰(R, T^*(1)) − [Q₁₁:Q₁₁]·rank T₁₁⁻`. The Weil pairing makes `det T = Z₁₁(1)`, on which complex conjugation acts by `−1`, so its eigenvalues on `T` are `+1, −1` and `rank H⁰(R,T^*(1)) = 1`. For the relaxed structure (`T₁₁⁺ = T`, `T₁₁⁻ = 0`) — the one whose Selmer complex is `RΓ(Z_S,T)` — `χ = 1 − 0 = 1`; for the ordinary Greenberg structure `χ = 1 − 1 = 0` (Remark 3.14's `SS₀`; Lemma 3.8's Heegner analogue).

Definition 2.16's formula (2.2.2) at the tree's own vertex `n = 397·991` on `A₁ = E[11]`: the three core-vertex conditions hold — `H¹_{𝓕^n}(Q,E[11])` is an `F₁₁`-space (free); `H¹_{/𝓕}(Q_ℓ,E[11]) = H¹_s(Q_ℓ,E[11])` is 1-dimensional at 397 and 991 (`E(Q_ℓ)[11] ≅ E(F_ℓ)[11] ≅ Z/11` since `11 ∥ #E(F_ℓ)`, so `dim H¹ = 2`, `dim H¹_f = 1`); and `H¹_{(𝓕^n)^*}(Q,E[11])` — strict at 11 and at both primes — sits inside `ker(loc_n on Sel₁₁(E/Q))`, which is 0 because `det M_loc = 2 ≠ 0` (RUN-022's `Sel_both = 0`; conditional on the inherited `Ш[11] = 0`). Then Proposition 2.18(i), `0 → H¹(Z_S,E[11]) → H¹_{𝓕^n} → ⊕_ℓ H¹_{/𝓕}(Q_ℓ) → Sel_str(E[11])^* → 0`, with `dim H¹(Z_S,E[11]) = 1 + dim Sel_str(E[11]) = 3 − ρ` and `dim Sel_str(E[11]) = 2 − ρ`, gives `dim H¹_{𝓕^n}(Q,E[11]) = 3` whatever `ρ`, and

```text
χ(𝓕_can) = dim H¹_{𝓕^{397·991}}(Q,E[11]) − (dim H¹_{/𝓕}(Q₃₉₇) + dim H¹_{/𝓕}(Q₉₉₁)) = 3 − (1 + 1) = 1.
```

The two formulas agree, so Hypothesis 2.17 (`χ(𝓕) = χ(𝓕)`) holds at `n = 1` with this vertex; the general existence of core vertices for `𝓕_can` under (sur) is cited (MR04, BSS19), not read. **The core rank is 1, not 2.** A core rank misread as 2, or a vertex dimension moved, is planted and caught (`dag-a-core-rank`).

## 5. `det⁻¹(C_Nek) ≅ SS_χ` as an actual map, every datum itemised

**Hypothesis 2.11** for the relaxed structure, each item finite: (i) `E(Q)[11^∞] = 0`; (ii) `T₁₁⁻ = 0`; (iii) `A₁₁⁻ = 0` and `H⁰(Q₁₁, T^∨(1)) = E(Q₁₁)[11^∞] = 0`; (iv) at 389 `H¹_ur = Z₁₁(1)/(389−1) = 0` and `H¹(Q₃₈₉,T) = 0`, equal. `T^{I₃₈₉} = Z₁₁(1)` is free. `RΓ_ur(Q₃₈₉,T)` and `RΓ(Q₃₈₉,T)` are both acyclic, so `RΓ_{/ur}(Q₃₈₉,T)` is, and with the `S_p`-term `RΓ(Q₁₁, T⁻) = RΓ(Q₁₁, 0) = 0`, Definition 2.2 gives `RΓ̃_{𝓕_can}(Q,T) ≃ RΓ(Z_S,T)`: CS's complex and MCS's complex are one object. `397, 991 ∈ 𝒫₁` (§3.1's condition `H¹_{/𝓕}(Q_ℓ,E[11]) ≅ F₁₁`), and `397·991` is a core vertex (§4).

| step | ring | complex | Selmer structure | lattice / module | hypotheses | status |
| --- | --- | --- | --- | --- | --- | --- |
| `𝔷_{Q∞}` | `Λ` | `RΓ(Z_S,𝐓)` | relaxed at 11 | `det⁻¹_Λ RΓ(Z_S,𝐓)` | Conjecture 2.2.2 (§2) | basis, CITED_THEOREM |
| `𝔷_{Q∞}(1)` | `Z₁₁` | `RΓ(Z_S,T) = RΓ̃_{𝓕_can}(Q,T)` | relaxed at 11 | `Z₁₁(P∧Q) ⊗ Sel_str(T)` | (2.14); the acyclicity above | basis, nonzero |
| `ε₁ = ϖ₁(𝔷_{Q∞}(1) mod 11)` | `F₁₁` (`n = 1`) | `C_{PT,n} = [H¹_{𝓕^n}(Q,E[11]) → ⊕_ℓ H¹_{/𝓕}(Q_ℓ,E[11])] ≃ C_Nek/11` (Theorem 2.20) | relaxed at 11 | `SS₁(E[11], 𝓕_can, 𝒫₁) ≅ det⁻¹(C_PT) = ∧³H¹_{𝓕^n} ⊗ H¹_s(Q₃₉₇)^* ⊗ H¹_s(Q₉₉₁)^*`, free of rank one (Prop 3.3); degree `3 = χ + ν(n)`; the duals are `W_ℓ = E(F_ℓ)⊗F₁₁` by local Tate duality | Hyp 2.11 (i)–(iv); Hyp 2.17 at `n = 1`; Theorem 3.4 itself cited | basis, nonzero |

`Δ_A := ε₁` (in characteristic zero, `𝔷_{Q∞}(1)`) is nonzero because a basis of a free rank-one module is. Corollary 3.5: `Fitt^i_{F₁₁}(Sel_str(E[11])^*) = I_i(ε₁)`, so `I₀ = 0`, `I₁ = F₁₁` — content about the **strict** Selmer group, the dual of the relaxed structure, not about the rank-2 Selmer group. The Selmer structure is "relaxed at 11" at every step; composing the relaxed basis into the ordinary `SS₀` is planted and caught (`dag-a-stark`), and so is identifying `SS₁` with `H¹(Z_S,𝐓)` or `Λ` because all have rank one — rank one is a property, not an isomorphism; the only identifications used are (2.9), (2.14), Proposition 3.3 and Theorem 3.4, each a named canonical map.

**The ordinary channel, read not composed.** Remark 3.14: Mazur's main conjecture ⟺ a unique `𝔷_∞^MSD ↦ 𝓛^MSD`, `ε^MSD ∈ SS₀`, `Fitt^i(Sel_{11^∞}(E/Q)^∨) = Θ_i^MSD`. Under the inherited input `Sel_{11^∞}(E/Q)^∨ ≅ Z₁₁²`, so `Θ₀ = Θ₁ = 0` and `Θ₂ = Z₁₁`; RUN-068's `δ̃_{397·991} = 5 ≠ 0` is a level-2 witness consistent with that. The bridge between the channels is Kim 3.11 / the Coleman map (`exp^* ∘ loc_p^s`), up to the unit `u` — which is why `u` must not be mistaken for a characteristic-zero scalar.

**A-mod11-compat.** Stark line at `n`: `∧³H¹_{𝓕^n}(Q,E[11]) ⊗ W₃₉₇ ⊗ W₉₉₁`. `B_N`'s line (RUN-103): `(∧²Sel₁₁)⁻¹ ⊗ W₃₉₇ ⊗ W₉₉₁`. Shared: the local factor. Different: the global factor, degree 3 on the relaxed-at-`n` Selmer group against degree `−2` on the Selmer group. What `det(loc_n) ≠ 0` does is certify the core vertex — the very thing that makes `SS₁` at `n` a single line. An equality of elements between the two lines is UNMEASURED, and by the instruction not necessary for A; judging A FAIL for its absence is planted and caught (`dag-a-char0-verdict`).

## 6. The ten mutations and two controls

| mutation | check |
| --- | --- |
| ideal equality (2.12) read as producing the distinguished element by itself | `dag-a-objects` |
| a canonical isomorphism read as making an arbitrary element canonical | `dag-a-objects` |
| core rank misread as 2 | `dag-a-core-rank` |
| inconsistent Selmer structures composed (relaxed basis into `SS₀`) | `dag-a-stark` |
| A FAIL for lack of a mod-11 equality | `dag-a-char0-verdict` |
| mod-11 witness constructing the characteristic-zero element backwards | `dag-a-placement` |
| main-conjecture theorem applied over a failing hypothesis | `dag-a-imc` |
| two explicit vectors demanded before PASS | `dag-a-char0-verdict` |
| merge on rank one | `dag-a-stark` |
| Kim's `u` taken as the characteristic-zero scalar | `dag-a-placement` |

Controls: **A** — the perfect complex `[Z² → Z]` with row `(2, 3)`, standard bases; the canonical basis `e₁∧e₂ ⊗ f⁻¹` of `det⁻¹(C)` is carried by the canonical isomorphism to `det H⁰ = Z·(3,−2)` onto `±(3,−2)`, a generator (the lift `s = (−1, 1)` has `det[k; s] = 1`): accepted at once, no explicit vector demanded. **B** — only the equality of characteristic ideals `(6) = (6)`: the generators are `±6`, none distinguished; PASS refused. Each is also planted in its refused / accepted form (`dag-a-controls`). Also planted: `a₃₈₉` declared `−1`; `a₁₁` declared `0`; the strict slope declared 2; the index declared 11; the ranks declared `(3, 2)`; the core ranks `(2, 0)`; `dim H¹_{𝓕^n}` declared 4; control A's row `(2, 4)` (not surjective); `Q` replaced by a point off the curve. Five drill controls — the multiplier 32 or 48 at 11, control A's row `(3, 5)` or `(5, 7)`, control B's ideal `(10)` — must stay green.

## 7. What this round does not claim

* the value of the unit `ε` (`𝔷_{Q∞}(1) = ε·(P∧Q) ⊗ R_str`) — not computed; `ε₁ = [det loc_{397·991}]` or any equality between the Stark line and `B_N`'s line;
* the main conjecture re-proved here (it is cited: Wan15, BCS25, Kato); `Ш(E/Q)[11] = 0` verified here (it is inherited); the internal hypotheses of Wan15, BCS25, Kato 12.4(1), or the MR04/BSS19 existence of core vertices in general; the Manin constant or optimality;
* a rank-2 Stark module — the core rank is 1, and the instruction's "do not force" was followed;
* anything about B, C, D, E, F, G; BSD. What the placement `Z₁₁(P∧Q) ⊗ Sel_str(T)` suggests for C (a trivialisation of the strict line is a height-type datum: MCS §4 is about derived heights) is noted, not opened.

## 8. Drill

The whole-tree mutation drill: **622 planted defects, 622 caught by the check named for them, 0 uncaught, 0 caught by the wrong check, 164 controls undisturbed, state restored, over 193 checks** (first run, as 8 shards). Twenty-one of the defects and five of the controls are this round's. The report and package were written before the drill was launched and nothing was touched while it ran.

## 9. Status

```text
normalization certificate C1–C10   CLOSED (architecture), frozen
DAG   A  PASS by cited theorem — 𝔷_{Q∞} exists, unique, a basis; 𝔷_{Q∞}(1) ∈ Z₁₁(P∧Q) ⊗ Sel_str(T); core rank 1; ε₁ a basis of SS₁;
         A-mod11-compat UNMEASURED (not necessary); scalar ε UNMEASURED
      B–G  untouched
BSD   OPEN
```